Wait, stay with me here, she lost all her bending right? Except for Airbending. The first book is Air, I'm assuming the next 3 seasons will be about her RElearning fire, earth and water. Just like Aang.

"Here’s the thing: Did she actually get her bending back? Did we see her bend outside of the Avatar State? No, we didn’t. Aang regrew the arm she lost. She might still have some rehab to go through (that is all hypothetical at this point, though). You might argue that Lin Beifong got her powers back when Korra re-bended her, and I’ll argue that bending works differently for the Avatar than it does for regular benders. What I mean to say is that there is the possibility that she still needs to relearn the three other element bending."

The Avatar state doesn't always mean that they can bend all elements anyway. It implies that she did get the other elements back. although they may not be mastered. An example of this would be episodes 118-120: "The Waterbending Master" and "The Siege of the North" Parts 1 and 2. In here, Aang activated his avatar state and could only bend water. There are two explanations for this. 1) The creators didn't want to show him bend the other elements to keep the story up longer (i.e. traveling the world and learning the elements) or 2) He couldn't because he hasn't learned it yet.

From the wikia: "While in this state, an Avatar has access to bending techniques they themselves may have yet to learn..." It says nowhere that they can use other types of bending. It just says techniques that they have yet to learn; initially being able to waterbend a simple stream to being able to move large bodies of water with ease.